Properties

Label 4732.2.g.d
Level $4732$
Weight $2$
Character orbit 4732.g
Analytic conductor $37.785$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4732,2,Mod(337,4732)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4732, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4732.337");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4732 = 2^{2} \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4732.g (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(37.7852102365\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 364)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 3 i q^{5} + i q^{7} - 3 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 3 i q^{5} + i q^{7} - 3 q^{9} - 2 i q^{11} + 7 q^{17} - 2 i q^{19} + 4 q^{23} - 4 q^{25} + q^{29} - 4 i q^{31} + 3 q^{35} + i q^{37} + 3 i q^{41} + 6 q^{43} + 9 i q^{45} - 10 i q^{47} - q^{49} - 7 q^{53} - 6 q^{55} + 6 i q^{59} + 7 q^{61} - 3 i q^{63} - 8 i q^{67} + 6 i q^{71} - 11 i q^{73} + 2 q^{77} - 14 q^{79} + 9 q^{81} + 14 i q^{83} - 21 i q^{85} - 10 i q^{89} - 6 q^{95} - 2 i q^{97} + 6 i q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{9} + 14 q^{17} + 8 q^{23} - 8 q^{25} + 2 q^{29} + 6 q^{35} + 12 q^{43} - 2 q^{49} - 14 q^{53} - 12 q^{55} + 14 q^{61} + 4 q^{77} - 28 q^{79} + 18 q^{81} - 12 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4732\mathbb{Z}\right)^\times\).

\(n\) \(2367\) \(2705\) \(4565\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
337.1
1.00000i
1.00000i
0 0 0 3.00000i 0 1.00000i 0 −3.00000 0
337.2 0 0 0 3.00000i 0 1.00000i 0 −3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4732.2.g.d 2
13.b even 2 1 inner 4732.2.g.d 2
13.d odd 4 1 4732.2.a.b 1
13.d odd 4 1 4732.2.a.f 1
13.f odd 12 2 364.2.k.b 2
39.k even 12 2 3276.2.z.a 2
52.l even 12 2 1456.2.s.d 2
91.w even 12 2 2548.2.l.f 2
91.x odd 12 2 2548.2.i.c 2
91.ba even 12 2 2548.2.i.f 2
91.bc even 12 2 2548.2.k.b 2
91.bd odd 12 2 2548.2.l.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
364.2.k.b 2 13.f odd 12 2
1456.2.s.d 2 52.l even 12 2
2548.2.i.c 2 91.x odd 12 2
2548.2.i.f 2 91.ba even 12 2
2548.2.k.b 2 91.bc even 12 2
2548.2.l.c 2 91.bd odd 12 2
2548.2.l.f 2 91.w even 12 2
3276.2.z.a 2 39.k even 12 2
4732.2.a.b 1 13.d odd 4 1
4732.2.a.f 1 13.d odd 4 1
4732.2.g.d 2 1.a even 1 1 trivial
4732.2.g.d 2 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(4732, [\chi])\):

\( T_{3} \) Copy content Toggle raw display
\( T_{5}^{2} + 9 \) Copy content Toggle raw display
\( T_{17} - 7 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 9 \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( T^{2} + 4 \) Copy content Toggle raw display
$13$ \( T^{2} \) Copy content Toggle raw display
$17$ \( (T - 7)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + 4 \) Copy content Toggle raw display
$23$ \( (T - 4)^{2} \) Copy content Toggle raw display
$29$ \( (T - 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 16 \) Copy content Toggle raw display
$37$ \( T^{2} + 1 \) Copy content Toggle raw display
$41$ \( T^{2} + 9 \) Copy content Toggle raw display
$43$ \( (T - 6)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 100 \) Copy content Toggle raw display
$53$ \( (T + 7)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 36 \) Copy content Toggle raw display
$61$ \( (T - 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 64 \) Copy content Toggle raw display
$71$ \( T^{2} + 36 \) Copy content Toggle raw display
$73$ \( T^{2} + 121 \) Copy content Toggle raw display
$79$ \( (T + 14)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 196 \) Copy content Toggle raw display
$89$ \( T^{2} + 100 \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
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